L10a152

L10a151

L10a153

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Link Presentations

[edit Notes on L10a152's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,14,15,13 X16,8,17,7 X8,16,9,15 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {5, -4, 8, -6, 7, -3}, {9, -1, 4, -5, 10, -2, 6, -7, 3, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-9q^{4}+12q^{3}+q^{-3}-12q^{2}-2q^{-2}+11q+6q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-6}+a^{-6}-2z^{4}a^{-4}-3z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-4a^{-2}-2z^{4}-4z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-9z^{4}a^{-6}+8z^{2}a^{-6}-3a^{-6}+6z^{7}a^{-5}-6z^{5}a^{-5}+z^{3}a^{-5}+za^{-5}+4z^{8}a^{-4}-7z^{4}a^{-4}+3z^{2}a^{-4}-a^{-4}z^{-2}+2a^{-4}+z^{9}a^{-3}+10z^{7}a^{-3}-27z^{5}a^{-3}+24z^{3}a^{-3}-10za^{-3}+2a^{-3}z^{-1}+7z^{8}a^{-2}+a^{2}z^{6}-15z^{6}a^{-2}-4a^{2}z^{4}+15z^{4}a^{-2}+5a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}+10a^{-2}+z^{9}a^{-1}+2az^{7}+6z^{7}a^{-1}-5az^{5}-23z^{5}a^{-1}+2az^{3}+23z^{3}a^{-1}+az-10za^{-1}+2a^{-1}z^{-1}+3z^{8}-8z^{6}+8z^{4}-7z^{2}-z^{-2}+4$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   3 1 -2 11                 4     4 9               5 3     -2 7             7 4       3 5           6 6         0 3         5 6           -1 1       4 7             3 -1     2 4               -2 -3     4                 4 -5 1 2                   -1 -7 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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